Suppose four groups of students were randomly assigned to be taught with four different techniques, and their achievement test scores were recorded. Are the distributions of test scores the same, or do they differ in location? The data is presented in the table below.

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Suppose four groups of students are randomly assigned to be taught with four different techniques, and their achievement test scores are recorded. Are the distributions of test scores the same, or do they differ in location? The data is presented in the table below.

Motivational Example

Suppose four groups of students are randomly assigned to be taught with four different techniques, and their achievement test scores are recorded. Are the distributions of test scores the same, or do they differ in location? The data is presented in the table below.

Teaching Method

Method 1

Method 2

Method 3

Method 4

Index

65

75

59

94

87

69

78

89

73

83

67

80

79

81

62

88

The small sample sizes and the lack of distribution information of each sample illustrate how ANOVA may not be appropriate for analyzing these types of data.

The Kruskal-Wallis Test

Kruskal-Wallis One-Way Analysis of Variance by ranks is a non-parametric method for testing equality of two or more population medians. Intuitively, it is identical to a One-Way Analysis of Variance with the raw data (observed measurements) replaced by their ranks.

Since it is a non-parametric method, the Kruskal-Wallis Test does not assume a normal population, unlike the analogous one-way ANOVA. However, the test does assume identically-shaped distributions for all groups, except for any difference in their centers (e.g., medians).

Calculations

Let N be the total number of observations, then .

Let R(Xij) denotes the rank assigned to Xij and let Ri be the sum of ranks assigned to the ith sample.

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The SOCR program computes Ri for each sample. The test statistic is defined for the following formulation of hypotheses:

Ho: All of the k population distribution functions are identical.

H1: At least one of the populations tends to yield larger observations than at least one of the other populations.

Suppose {} represents the values of the ith sample, where .

Test statistics:

T = ,

where

.

Note: If there are no ties, then the test statistic is reduced to:

.

However, the SOCR implementation allows for the possibility of having ties; so it uses the non-simplified, exact method of computation.

Multiple comparisons have to be done here. For each pair of groups, the following is computed and printed at the Result Panel.

.

The SOCR computation employs the exact method instead of the approximate one (Conover 1980), since computation is easy and fast to implement and the exact method is somewhat more accurate.